Simulating the path for Interest Rate

Question

There are many ways to short term rates like Ho-lee process, HW process. However I failed to understand how this information can be used to simulate the process for Overnight rate like EONIA etc.

Can you please refer to any online technical papers to simulate such Overnight process?

Just to explain more, let say I define a HW process as follows

$dr(t) = (\theta(t)-\alpha r(t))dt+\sigma dW_t$

With this process, I can estimate a discount bond as $P(t, t+1)$ based on the estimated parameters (refer to https://en.wikipedia.org/wiki/Hull–White_model)

So is it right to day that the process for OI rate is just the process for $P(t,t+1)$?

Is there any software implementation like R/Python that someone can please refer to?

Thank you very much.

Answer 1

You are on the right path but here $P(t,t+1)$ is not your overnight rate but a daily discount factor. To convert to a simply compounded daily rate, which would be your overnight rate, you would do something like this $$ R(t,t+1) = (1 - P(t,t+1))/(\tau P(t,t+1)) $$

Here $\tau$ is your daycount fraction for $1$ day

As I mentioned in the comment $R(t,t+1)$ is very close to the short rate $r(t)$ and, depending on the intended application, you might want to use the latter as it is a bit simpler. Otherwise use $R$